How to find group homomorphisms from one group to another

Question

I am trying to figure out all the homomorphisms from $\mathbb{Z}_2\times\mathbb{Z}_2$ to $\mathbb{Z}_2$.

Is there a good process for doing such a think? I am getting lost...

Answer 1

Look at the possible kernels. What are the normal subgroups of $G=\mathbb{Z}_2\times \mathbb{Z}_2$?

$1$ and $G$ are always normal subgroups of any group $G$. There are three proper subgroups, generated by $(1,0)$, $(0,1)$, and $(1,1)$, respectively, and all are normal because $\mathbb{Z}_2\times\mathbb{Z}_2$ is abelian. Let's call those $K_1,K_2,$ and $K_3$, respectively.

What quotient groups do they yield?

For each $i$, $\mathbb{Z}_2\times\mathbb{Z}_2/K_i \cong \mathbb{Z}_2$ by simple order arguments. Of course, for any group $G$, $G/G$ is the trivial group, and $G/1=G$.

Which of these groups could fit into $\mathbb{Z}_2$ as images?

The $G/K_i$ and the $G/G$ do, and $G/1$ doesn't.

Now use the first isomorphism theorem to finish.